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Symmetric Markov chains in ℤd: How fast can they move?
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  • Published: June 1989

Symmetric Markov chains in ℤd: How fast can they move?

  • Martin T. Barlow1 &
  • Edwin A. Perkins2 

Probability Theory and Related Fields volume 82, pages 95–108 (1989)Cite this article

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  • 17 Citations

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Summary

Consider a reversible Markov chain X n which takes values in a subset of ℤd. If the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition, Varopoulos, Carne and Kesten have obtained estimates on \(P(|X_n - X_{\text{0}} | > \lambda n^{1/2} )\) which exhibit a Gaussian tail in λ but blow up as n→∞. Following Kesten's approach we derive bounds which are uniform in n in some special cases. Our main result, however, is an example which shows that in general the estimates of Varopoulos, Carne and Kesten are essentially best possible.

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References

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Authors and Affiliations

  1. Statistical Laboratory, 16 Mill Lane, CB2 1SB, Cambridge, U.K.

    Martin T. Barlow

  2. Mathematics Department, University of British Columbia, V6T 1Y4, Vancouver, B.C., Canada

    Edwin A. Perkins

Authors
  1. Martin T. Barlow
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  2. Edwin A. Perkins
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Additional information

Research partially supported by an S.E.R.C. (U.K.) visiting fellowship and an operating grant from N.S.E.R.C. of Canada

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Barlow, M.T., Perkins, E.A. Symmetric Markov chains in ℤd: How fast can they move?. Probab. Th. Rel. Fields 82, 95–108 (1989). https://doi.org/10.1007/BF00340013

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  • Received: 23 March 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00340013

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Keywords

  • Markov Chain
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Invariant Measure
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